A Note Regarding Second-Order $\Gamma$-limits for the Cahn--Hilliard Functional

TL;DR

This paper fully characterizes the second-order $ ext{Gamma}$-limits of the mass-constrained Cahn--Hilliard functional, removing a key assumption from previous work and advancing the understanding of its asymptotic behavior.

Contribution

It demonstrates that the critical assumption in prior second-order $ ext{Gamma}$-convergence results for the Cahn--Hilliard functional is unnecessary, completing the asymptotic analysis.

Findings

Complete second-order $ ext{Gamma}$-limit characterization.

Removal of a key assumption from previous proofs.

Clarification of the asymptotic development of the functional.

Abstract

This note completely resolves the asymptotic development of order $2$ by $\Gamma$-convergence of the mass-constrained Cahn--Hilliard functional, by showing that one of the critical assumptions of the authors' previous work (Leoni, Murray, Second-order $\Gamma$-limit for the Cahn--Hilliard functional, Arch. Ration. Mech. Anal. 219, 3, 2016) is unnecessary.